Fluid Mechanics - Find the required power

[spatual]

Hi I don't know where to start with this problem. Any help would be greatly appreciated.

A cone is rotating in a slot at constant angular speed w. The gap h separating the cone and the lsot's wall is much smaller than the cone's height b. A fluid with viscosity mu fills the gap. Obtain an expression for the power required to just overcome the viscous resistance. Calculate this power for a rotating speed of 500rpm, h = .2 mm, b= 10cm mu= .025 Pa.s cone's half angle alpha = 25 degrees

Because the example given uses the formulas below but it was for a cylinder and velocity was in m/s. As a result I have no idea what to do.
Shear stress= mu * (change in velocity/gap between cones)
Shear force = shear stress * area (surface?)
Power = Force * Velocity

I know I have to convert rpm to radians and that as the radius changes the velocity changes as well.

As height b changes so does the radius (which is the lever arm?) but I am not sure how to express that. Any pointers on how to get started would be great.

Thanks.

 

[Valkarie]

To calculate the power required to just overcome the viscous resistance, you need to first calculate the shear force between the cone and the slot wall. The shear force is equal to the product of the shear stress and the area of the cone surface exposed to the slot wall. The shear stress is equal to the viscosity of the fluid multiplied by the change in velocity between the cone and the slot wall, divided by the gap h between them. The area of the cone surface exposed to the slot wall can be calculated using the formula A = pi * r² * (1 - cos(alpha)), where alpha is the half-angle of the cone, r is the radius of the cone, and pi is the constant 3.14.Once you have the shear force, you can calculate the power required to overcome the viscous resistance by multiplying the shear force by the velocity of the cone. To convert the rotational speed in rpm to radians per second, divide the rpm by 60 and multiply the result by 2 π.For the example given, with w = 500rpm, h = 0.2mm, b = 10cm, μ = 0.025 Pas, and α = 25°, the power required is:Shear Stress = μ * (w / h)= 0.025 * (500 / 0.2)= 625 PaArea of cone surface exposed to slot wall = π * (b/2)² * (1 - cos(25°))= 3.14 * (5)² * (1 - 0.906)= 44.47 cm²Shear Force = Shear Stress * Area = 625 * 44.47= 27773.75 NPower = Shear Force * Velocity = 27773.75 * (500/60 * 2π)= 3265.4 W

 

[Mene]

Hello, thank you for reaching out for help with this problem. I can definitely assist you in finding the required power in this fluid mechanics scenario.

First, we need to understand the concept of power in fluid mechanics. Power is the rate at which work is done or energy is transferred. In this case, the power required is the rate at which the fluid is being moved or transferred through the gap between the rotating cone and the slot's wall.

To find the power required, we need to consider the forces acting on the fluid in the gap. The main force acting on the fluid is the viscous resistance, which is caused by the friction between the fluid and the cone's surface. This force is directly proportional to the surface area of the cone and the viscosity of the fluid.

Using the given formula for shear stress, we can express the shear force acting on the fluid as follows:

Shear force = mu * (change in velocity/gap between cones) * surface area

Since the cone is rotating at a constant angular speed, the velocity at any point on the surface of the cone can be expressed as v = w*r, where w is the angular speed in radians per second and r is the distance from the center of rotation to the point on the surface.

Now, we need to consider the geometry of the cone and the gap. Since the gap is much smaller than the cone's height, we can approximate the cone as a cylinder with a height of b and a radius of r = b*tan(alpha), where alpha is the cone's half angle.

The surface area of the cone can be expressed as A = 2*pi*r*b. Substituting this into the equation for shear force, we get:

Shear force = mu * (w*b*tan(alpha)/h) * 2*pi*r*b

To find the power required, we need to multiply this shear force by the velocity of the fluid. Since the velocity of the fluid is the same as the velocity of the cone's surface, we can use the expression v = w*r to find the velocity. Substituting this into the equation for power, we get:

Power = (mu * (w*b*tan(alpha)/h) * 2*pi*r*b) * (w*r)

Now, we can plug in the given values for w, h, b, mu, and alpha to solve for the required power. Remember to convert the angular speed from rpm to radians per second before